TDDFT Explorations in Phase-Space. Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York

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1 TDDFT Explorations in Phase-Space Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York

2 TDDFT in Phase-Space Outline Why phase-space DFT? Some motivating examples and initial explorations. Less memory in phase-space DFT: Hooke s atom example Autoionizing Resonances in TDDFT The structure of the kernel needed for those arising from a bound double excitation

3 Two types of challenges in TDDFT: Why Phase-space DFT? --- (1) In several situations, v xc [n;ψ 0,Φ 0 ](r,t) is difficult (unnatural?) to approximate well. eg. strong fields memory-dependence often important eg. some quantum control problems

4 Example: Some Quantum Control problems Consider pumping He from ground (1s 2 ) to first accessible excited state (1s2p). Problem!! The KS state remains doubly-occupied throughout cannot evolve into a singly-excited KS state. Simple model: evolve two electrons in a harmonic potential from ground-state (KS doubly-occupied φ 0 ) to the first excited state (φ 0,φ 1 ) : TDKS -- KS system achieves the target excited-state density, but with a doublyoccupied ground-state orbital!! -- Yet this is how exact TDDFT describes the dynamics the exact v xc is unnatural and difficult to approximate. Maitra, Burke, Woodward PRL (2002)

5 Two types of challenges in TDDFT: Why Phase-space DFT? --- (1) In several situations, v xc [n;ψ 0,Φ 0 ](r,t) is difficult (unnatural?) to approximate well. eg. strong fields memory-dependence often important eg. some quantum control problems --- (2) Need additional observable functionals when the observable of interest is often not directly obtainable from Φ s eg. Kinetic energies (ATI spectra (Ngyuen, Bandrauk, Ullrich PRA (2004), or ion-momentum distributions Wilken and Bauer (PRA 2007) ) eg. pair density for double-ionization yields (but see Wilken & Bauer PRL (2006) )

6 Example: Ion-recoil in Non-sequential double ionization Progress has been made for approxs in TDDFT to get accurate yields for NSDI. [Lein & Kuemmel PRL 94, (2005); Wilken & Bauer PRL 97, (2006) ] But what about momentum distributions? Ion-recoil p-distributions computed from exact KS orbitals are poor, eg. (Wilken and Bauer, PRA 76, (2007)) Wilken and Bauer derive a product-phase approximation But generally KS momentum distributions don t have anything to do with the true p-distribution ( in principle the true p-dist is a functional of the KS system but what functional?!)

7 w(r,p,t) Idea: Explore a generalized DFT based on Phase-Space (quasiprobability) density Basic variable has more information functionals may be simpler to approximate More observables directly obtained without additional observable-functionals eg.,momentum, kinetic energy

8 Wigner function 1-body density matrix (1-DM) contains both r and p information So, in this TD-phase-space-DFT, would need no additional observable functionals for the expectation value of any one-body operator Literature: Wigner function mostly used for one-particle problems, esp. to make classical quantum analogies. However some work on many-electron problems, particularly for transport, eg. Jacoboni, Rep. Prog. Phys (2004); Cancellieri, Bordone, Bertoni, Ferrari & Jacoboni,J. Comp. Elec. 3, 411 (2005)

9 Simple example: 2-el 1-d Model of Ionization n(x) true n(p) KS n(p) True W(r,p) KS W(r,p) KS phase-space and momentum distributions can be significantly different from true in ionization

10 TDKS Momentum distributions Wigner phase-space distributions

11 Equivalence of the Wigner fn and the 1-body Density Matrix w and ρ 1 contain the exact same information, just in a different form. For some purposes (eg eqns) ρ 1 is easier to use; other purposes (eg visualization, semiclassical analyses, possibly approximations??), w easier. TD-DMFT already some work in linear response regime: Pernal, Gritsenko, Baerends Phys. Rev. A 75, (2007), Giesbertz, Baerends, Gritsenko Phys. Rev. Lett. 101, (2008) Also note: Ground-state DMFT: Lathiotakis, Helbig & Gross PRA (2005), Buijse & Baerends, Mol. Phys. (2001), Sharma et al., Helbig et al. recent papers on the arxiv

12 Equation of motion: First, note no KS system, since the interacting ρ 1 is not idempotent and any non-interacting ρ 1 must be. So we deal with ρ 1 (or w) directly with it s equation of motion: kinetic energy done exactly (unlike in TDDFT) + We d like these to: Capture correlation Change occupation # s of TDNO (elec qm control) BBGKY truncation, but. Semiclassical? Want this in terms of ρ 1 Approximations needed here (see also earlier references, adiabatic Giesbertz et al) see also Appel & Gross, arxiiv (2008)

13 So, we have a lot of work to do to develop such functionals! But, can we say anything about their memory-dependence? TDDFT Memory arises because of the reduced description of the system in terms of density alone. So, could functionals of the phase-space density F[w] be less nonlocal in time? Revisit Hooke s atom example

14 TDDFT History-dependence in Hooke s atom Eg. Time-dependent Hooke s atom exactly numerically solvable 2 electrons in parabolic well, time-varying force constant parametrizes density k(t) = *cos(0.75 t) Any adiabatic (or even semi-local-in-time) approximation would incorrectly predict the same v c at both times. Time-slices where n(t) is locally and semi-locally identical but v c is quite distinct v c is generally a very non-local functional in time of the density Hessler, Maitra, Burke, (J. Chem. Phys, 117, )

15 Hooke s Atom Could functionals of the phase-space density F[w] be less non-local in time? Exploration: Look to see if w(r,p,t) distinguishes the system at times when n(r,t) is the same. To simplify: we actually look at and define:

16 t=12.3 t=20.3 At each pair of times: n(r,t) identical, v c (r,t) different. n c (p,t) different t=4.8 t=28.9 And, size of difference of n c (p,t) seems to scale directly with difference in v c( r,t) t=5.5 t=29.8 Instantaneous p-densities do distinguish differences in vc(rt) in cases where instantaneous n(r,t) are identical, i.e. functionals of w(r,p,t) may be less-non-local in time than those of n(r,t), so easier

approximate functional construction At least one example suggests functionals of w are less non-local in time than functionals of n

17 TD Phase-Space DFT: Summary so far Helps some observable problems -- eg. directly gives momentum distributions of ionized electrons in some region of space -- while those observables in turn may help with approximate functional construction At least one example suggests functionals of w are less non-local in time than functionals of n Next steps: -- can we prove a 1-1 mapping for the TD case for nonlocal potentials? -- derive functionals for w 2 (or ρ 2 ) in terms of w (or ρ 1 )?! (Rajam, Hessler, Gaun, & Maitra, in prep)

18 Autoionizing Resonances ( Fano s Face-Space)

19 Autoionizing Resonances When energy of a bound excitation lies in the continuum: KS (or another orbital) picture ω ω bound, localized excitation continuum excitation True system: Electron-interaction mixes these states Fano resonance ATDDFT gets these mixtures of single-ex s

20 Auto-ionizing Resonances in TDDFT Eg. Acetylene: G. Fronzoni, M. Stener, P. Decleva, Chem. Phys. 298, 141 (2004) But here s a resonance that ATDDFT misses: Why? It is due to a double excitation.

21 a i ω = 2(ε a ε i ) ω bound, localized double excitation with energy in the continuum single excitation to continuum Electron-interaction mixes these states Fano resonance ATDDFT does not get these double-ex

22 Photo-absorption cross section: The cross-section in TDDFT Where, ^ ^ How does the exact kernel add the resonant bump to χ s? Do a degenerate perturbation theory analysis in the continuum, diagonalizing the bound double-excitation with the continuum states. (c.f. bound-state doubles of Maitra, Zhang, Cave, Burke, JCP 120, 5932 (2004) Aha! Luckily this is closely along the lines of what Fano did in 1961

23 Near a Fano resonance Fano s Universal Resonance Formula ^ U. Fano, Phys. Rev. 124, 1866 (1961) ^ ^ Take T = n(r) where ^ ^ ^ ^ ^

24 The expression for q looks complicated but for our case, it is just 1!! ^ ^ because the doubleexcitation does not contribute to the KS cross-section, and the oscillator strength sum rule must be conserved, line-shape factor must be purely antisymmetric. q=1 Figure from U. Fano, Phys. Rev. 124,1866 (1961) ε

25 Consider Fano into TDDFT to be Kohn Sham states Adapt Fano analysis: For near resonance In this approx, we considered coupling between doubly-excited to continuum only, and took Γ as the smallest energy scale in the system.

26 To find we need Use Kramers-Kronig relation.... Assuming

27 So near the resonance looks like Re Im

28 Autoionizing Resonances: Summary Certain AI resonances cannot be captured by Adiabatic TDDFT as they arise from a doubly-excited bound-state. Derived the exact form for the frequency-dependent kernel that is needed, in the limit of a sharp and isolated resonance. Need to test on model and real systems! Krueger, Mullady, Maitra, in progress

29 Muchos Gracias à: Angel, Miguel, Hardy, Fernando & Hunter College Gender Equity Project Postdoc Opening And to you all for listening!

Advanced TDDFT II. Double Excitations in TDDFT

Advanced TDDFT II. Double Excitations in TDDFT f xc Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York First, quick recall of how we get excitations in TDDFT: Linear